let A, B, C, D, E, F, J, M be set ; :: thesis: for h being Function
for A9, B9, C9, D9, E9, F9, J9, M9 being set st h = (((((((B .--> B9) +* (C .--> C9)) +* (D .--> D9)) +* (E .--> E9)) +* (F .--> F9)) +* (J .--> J9)) +* (M .--> M9)) +* (A .--> A9) holds
rng h = {(h . A),(h . B),(h . C),(h . D),(h . E),(h . F),(h . J),(h . M)}
let h be Function; :: thesis: for A9, B9, C9, D9, E9, F9, J9, M9 being set st h = (((((((B .--> B9) +* (C .--> C9)) +* (D .--> D9)) +* (E .--> E9)) +* (F .--> F9)) +* (J .--> J9)) +* (M .--> M9)) +* (A .--> A9) holds
rng h = {(h . A),(h . B),(h . C),(h . D),(h . E),(h . F),(h . J),(h . M)}
let A9, B9, C9, D9, E9, F9, J9, M9 be set ; :: thesis: ( h = (((((((B .--> B9) +* (C .--> C9)) +* (D .--> D9)) +* (E .--> E9)) +* (F .--> F9)) +* (J .--> J9)) +* (M .--> M9)) +* (A .--> A9) implies rng h = {(h . A),(h . B),(h . C),(h . D),(h . E),(h . F),(h . J),(h . M)} )
assume h = (((((((B .--> B9) +* (C .--> C9)) +* (D .--> D9)) +* (E .--> E9)) +* (F .--> F9)) +* (J .--> J9)) +* (M .--> M9)) +* (A .--> A9) ; :: thesis: rng h = {(h . A),(h . B),(h . C),(h . D),(h . E),(h . F),(h . J),(h . M)}
then A1: dom h = {A,B,C,D,E,F,J,M} by Th63;
then B in dom h by ENUMSET1:def 6;
then A2: h . B in rng h by FUNCT_1:def 3;
M in dom h by A1, ENUMSET1:def 6;
then A3: h . M in rng h by FUNCT_1:def 3;
J in dom h by A1, ENUMSET1:def 6;
then A4: h . J in rng h by FUNCT_1:def 3;
F in dom h by A1, ENUMSET1:def 6;
then A5: h . F in rng h by FUNCT_1:def 3;
E in dom h by A1, ENUMSET1:def 6;
then A6: h . E in rng h by FUNCT_1:def 3;
A7: rng h c= {(h . A),(h . B),(h . C),(h . D),(h . E),(h . F),(h . J),(h . M)}

proof

let t be object ; :: according to TARSKI:def 3 :: thesis: ( not t in rng h or t in {(h . A),(h . B),(h . C),(h . D),(h . E),(h . F),(h . J),(h . M)} )
assume t in rng h ; :: thesis: t in {(h . A),(h . B),(h . C),(h . D),(h . E),(h . F),(h . J),(h . M)}
then consider x1 being object such that
A8: x1 in dom h and
A9: t = h . x1 by FUNCT_1:def 3;
( x1 = A or x1 = B or x1 = C or x1 = D or x1 = E or x1 = F or x1 = J or x1 = M ) by A1, A8, ENUMSET1:def 6;
hence t in {(h . A),(h . B),(h . C),(h . D),(h . E),(h . F),(h . J),(h . M)} by A9, ENUMSET1:def 6; :: thesis: verum

end;

D in dom h by A1, ENUMSET1:def 6;
then A10: h . D in rng h by FUNCT_1:def 3;
C in dom h by A1, ENUMSET1:def 6;
then A11: h . C in rng h by FUNCT_1:def 3;
A in dom h by A1, ENUMSET1:def 6;
then A12: h . A in rng h by FUNCT_1:def 3;
{(h . A),(h . B),(h . C),(h . D),(h . E),(h . F),(h . J),(h . M)} c= rng h by A12, A2, A11, A10, A6, A5, A4, A3, ENUMSET1:def 6;
hence rng h = {(h . A),(h . B),(h . C),(h . D),(h . E),(h . F),(h . J),(h . M)} by A7, XBOOLE_0:def 10; :: thesis: verum